Let \(R_p(I)\)  be the rotation poset of \(I\) . By [1]}, there is a 1-1 correspondence between closed subsets of \(R_p(I)\)  and the stable matchings of \(I\) .
Let \(S\)  be a maximum profile closed subset of the rotation poset \(R_p(I)\)  and let \(M\)  be the unique corresponding stable matching. Then, \(p(M) = p(M_0) + \sum _{\rho _i \in S} p(\rho _i)\) .
Suppose \(M\)  is not rank-maximal. Then there is a stable matching \(M^{\prime }\)  such that \(p(M^{\prime }) \succ p(M)\) . As above, \(M^{\prime }\)  corresponds to a unique closed subset \(S^{\prime }\)  of the rotation poset. Also \(p(M^{\prime }) = p(M_0) + \sum _{\rho _i \in S^{\prime }} p(\rho _i)\) . But \(p(M^{\prime }) \succ p(M)\)  and so \(S\)  cannot be a maximum profile closed subset of \(R_p(I)\) , a contradiction.
